Talk:Liouville's theorem (Hamiltonian)

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You're staring right at the equation! What do you mean state the theorem? Phys 19:52, 5 Sep 2003 (UTC)

Firstly, the equation I am staring at is something you have called a corollary, and that means it's not the theorem itself. Secondly, nothing is actually stated! You've left everything undefined, and no assumptions are given. If that's your idea of stating a theorem, you'll get straight Fs in any theoretical math course. The equation is a differential equation in which the notation { ρ, H } is not given any definition. "How probability distributions evolve under time evolutions"? What does that mean? One may assign to each instant t in time a probability distribution, say Pt in many ways, and there is no differential equation that this assignment must satisfy merely as a result of this thing's being an assignment of probability distributions to instants in time. Michael Hardy 23:23, 5 Sep 2003 (UTC)

... and is ρ supposed to be (1) a probability density function, or (2) a cumulative probability distribution function, or (3) a probability measure, or (4) something else? That's one thing you'd obviously need to state here in order to have a statement of the theorem. Michael Hardy 23:26, 5 Sep 2003 (UTC)

Yes, something odd here about the formulation, anyway. A smooth measure on a manifold has a different variance property to a function, no? So, is the point that in the Hamiltonian setting the background top-level differential form (wedge all the dp and dq) is invariant under the flow, meaning that we can blur the difference?

Charles Matthews 08:52, 10 Sep 2003 (UTC)

Restored this - my fault it went before. The page needs to be brought in line with symplectic topology, amongst other things.

Charles Matthews 20:09, 19 Sep 2003 (UTC)


I think the point here is that they want to talk about 'smooth measures' on the phase space, whether or not those are probability measures. 'Smooth measure' is a measure absolutely continuous with respect to the background measure μ, which is like

|dp^dq^...|

in usual notations. Then H means Hμ really, and the 'base case' H = 1 of the theorem is saying that the background Liouville measure μ is invariant under the flow. Well, I'm not saying this very well, but I think that's the setting.

Charles Matthews 16:47, 13 Sep 2004 (UTC)


Contents

[edit] help

Could someone please expand this article to help explain the theorem more fully. What is it used for? How is it used? What year did was it brought to light? How did it change or effect physics? Why is it significant? What are the practical ueses? Kingturtle 02:49, 4 Jun 2004 (UTC)

The information paradox article points here to find out why information is never destroyed. But I cannot remotely figure it out. This idea about information seems so important it would be nice to find a detailed explanation. Manuel

[edit] "distribution function"?

The article says:

In a system with Hamiltonian H and distribution function ρ, the theorem states that ...

Does "distribution function" mean "probability density function"? If so, it should say that, because the term distribution function is often taken to mean cumulative distribution function, which is of course a different thing. Michael Hardy 00:42, 12 Sep 2004 (UTC)

"Distribution function" seems to be the standard usage in the literature my job entails reading, even though the functions under consideration certainly aren't the cumulative distributions. For example, Blum and Rolandi's Particle Detection with Drift Chambers (Springer-Verlag, 1993), says the following, on p. 81:
The principal approximation of Sect. 2.2 was to take a single velocity c to represent the motion between collisions of the drifting electrons. In reality, these velocities are distributed around a mean value according to a distribution function
f0(c) dc
which represents the isotropic probability density of finding the electron in the three-dimensional velocity interval dcxdcydcz at c.
An interesting little paper by Blatt and Opie (J. Phys. A, Vol. 7, No. 15, 1997) derives the Boltzmann transport equation from Liouville's theorem, as given in this article. In the introduction, they say,
The distribution function is assumed to satisfy the Liouville equation
\frac{\partial \rho}{\partial t} = -[\rho,H_n]
where Hn is the N-particle Hamiltonian, the particles being confined to a rigid box of volume V, and the bracket on the right-hand side denotes a Poisson bracket.
"Distribution function" also seems to be the preference among my quantum and stat. mech. professors, but I'd have to dig out my lecture notes to be sure. What can I say? Physics people are strange. (But they lead charmed lives.)
Anville 15:43, 13 Sep 2004 (UTC)

This is one case where I'm inclined not to be tolerant of terminology differences between fields, and say simply that they've got it wrong. Michael Hardy 00:32, 19 Sep 2004 (UTC)

You can say that a single article got it wrong, but you can't say an entire field "got the terminology wrong" as there's no reason to prefer the terminology of one field over that of another. I can assure you that I've read dozens of articles in which this terminology is used. I would argue however, that what currently (before my edit) appears in the article is incorrect: What I call the quantum Liouville equation is certainly not THE master equation. Any linear 1st order differential equation is A master equation. Cederal 12:51, 17 April 2007 (UTC)

[edit] Anville's change 29/10/2004

Sorry I don't agree that determinism alone is sufficient to give the theorem!

The local density in phase space D=N/V. You need to argue that N is constant within the marked volume, and this I think is what your argument on non-crossing of trajectories does.

You also need to show that V doesn't change as you follow the test particle (total dV/dt =0).

I think in essence V doesn't change because the shrinkage of V in the q coordinate direction is exactly offset by an equivalent increase in the conjugate p direction, and this RELIES on Hamilton's relationships between the p's and q's and their time derivatives.


In addition, the relationship is between generalised coordinates and their conjugate momenta, so the use of q rather than x is usual.

I shall edit accordingly if you agree.

Linuxlad 17:20, 29 Oct 2004 (UTC)

To all who have corrected my mistakes, both major and minor:
Thank you. I completely agree with your objections, and I have very similar statements in my stat. mech notes. Why I didn't get things right the first time, I can't tell you—but hey, the article is much better now, so it's all okay in the end.
Gratefully, Anville 14:48, 27 Nov 2004 (UTC).


I think it would be nice if we could have a version of the proof which does not rely on Darboux coordinates, which only happens to be local in some symplectic manifolds. Phys 23:16, 11 Nov 2004 (UTC)


_____

[edit] Open note to Denevans

Can we have a discussion on your recent mods to this article? - whilst I'm sure many are useful, some of them seem to put the cart before the horse (if I were a 3rd year physicist I don't think I'd get past your early description in the phase-space section); and in one or two places seem to me to muddle things (eg in _my_ world the convective derivative is NOT a definition it's a deduction ; and again your statement on Hamiltonian and non-Hamiltonian systems is rather unclear).

Phase space (section 2) is in any case discussed elsewhere. Avogadro's number is an irrelevance - stat mechanics applies to much larger and smaller system's than the 1 gram-mol scale! (is this perhaps a typo?)

The equation (section 3) you call Liouville's appears to me be 'just' a conservation of D equation found by integrating system-point flows over a control volume (ie the RHS is div (D 'v') where 'v' is a p,q 'velocity') - it's NOT the equation (presently your second) which physicists and theoretical chemists look to to eg justify the weighting on the Boltzmann distro, - this has some real-world dynamics in it. This first equation is indeed used later on as the (nearly self-evident :-)) starting point for the 'fluids' proof of LT (this is not by any means original with me - I can point you to at least 2 similar proofs on the web, from Harvard and Oxford TP departments IIRC).

(In passing - Is there a subtle distinction between LE (which already has an article) and LT which we should strive to maintain?)

Can we avoid the use of the term symplectic system until later - it may well be idea du jour to mathematicians and advanced physicists - but Wiki still lacks an adequate succinct expostulation accessible to a graduand-physicist (of my generation.) At present all symplectic space refs spin off into ideas like 1-forms, tangent bundles etc., which may be simple but are not presently explained in terms accessible to those with only a standard 'maths-methods' background.

The dynamics of non-Hamiltonian systems and non-conservation of heat are of course important, but most physicists sidestep them, being interested in micro-systems, or treating the system as embedded in a larger system obeying conservation.

It may be that a dual approach combining the 'physics' and symplectic systems ways is doomed not to work, like the tensors fiasco, but we can at least try.


Can we talk? Your place or mine, or in the appropriate talk page? Linuxlad 15:14, 13 Feb 2005 (UTC)


[edit] Informal Proof & Symplectic Geometry equivalent

Having struggled through several hundred pages of Penrose (ref. 1) I find that the 'informal proof' (taught IIRC on Natural Sciences Tripos) is essentially the same as the symplectic geometry statement given in 20.4 ('Hamiltonian dynamics as symplectic geometry') of that book . Since one of the main problems for eg physicists is in reading across terms like '2-form', I have added a paragraph to 'guide' those who follow. Please feel free to amend or correct, but remember the intended audience for this section is from the physical sciences. Linuxlad 09:23, 8 Mar 2005 (UTC)

[edit] revision

I tried to revise this completely. There was a lot that was redundant on this page, and it wasn't ever clear where the main statement of the theorem was, or what it had to do with physics, so I tried to clean it up a little, and removed the excessive subsectioning. I don't think it is perfect, but I hope you'll agree its an improvement. –Joke137 22:47, 4 Jun 2005 (UTC)

My first impression is that it's mathematically top-heavy for my taste (Though I agree that it was in urgent need of a cleanup).

From a pedagogical point of view I think that there should a clear and early statement that the theorem is 0) that the phase space density around a state point (ie convective derivative) is constant, and that this follows from 1) if you follow a set of points, the stretch in dp is offset by the contraction in dq and/or 2) viewed as a fluid in phase space, the flow has zero divergence. The basic statement of the theorem and the key reasons should be accessible to a competent final-year physics or chemistry 'major' Linuxlad 09:40, 5 Jun 2005 (UTC)

Well, I tried to not introduce any unnecessary formalism, but of course a better writer could still simplify it. Liouville's theorem is a mathematical statement, so it is hard to imagine stating it in a non-mathematical way. I think that the statement (1) you make is the statement about volume, in particular the Jacobian of the time evolution, in "other statements." It seems no easier to formulate precisely than the usual way, in terms of Liouville's equation. In particular, I don't think that dq_i always compensates for dp_i: it could be that dp_j compensates, or even dq_j... (but see below - linuxlad)

(2) By zero divergence, I assume you mean in the static case. Otherwise the statement is that the total derivative of the flow vanishes. (see below - linuxlad) –Joke137 03:30, 6 Jun 2005 (UTC)

Later:- No, the effects must cancel in (p,q) pairs (because of the Hamilton relations - see my original para, which you've cut) - isn't this what you'ld expect? (the 'system' could comprise two pendulums at opposite ends of the universe, so p1,q1 and p2,q2 have no coupling) It's essentially the divergence statement (where the cancellation is also in p,q pairs) in a Lagrangian frame (as I think You're saying above), I think. The 'proof' is standard - I was given it in 3rd year physics many years ago, (and found it helpful and memorable) - and it's also on the web. :-) Linuxlad 08:01, 6 Jun 2005 (UTC)

That is only true for infinitesimal times. Of course in your example of seperated pendulums, it holds for infinite times, but for a general system there is mixing between all the variables. –Joke137 15:48, 9 Jun 2005 (UTC)

Incidentally I think you need care when using for single particles (as when you introduce Mr Vlasov) - since the theorem is about 'system points', and classically there could be any number of those, provided you choose an appropriate relative density.

_Your_ last step (using the Hamilton's relations) _is_ showing that the divergence is zero.

Linuxlad 08:01, 6 Jun 2005 (UTC)

I don't know exactly what you mean by "system points." The theorem is about a phase space distribution function, which could either, physically, represent one particle (with some uncertainty about its position in phase space) or several, non-interacting particles, as in the case of the Vlasov equation. This can then be generalized to the Boltzmann equation, in which a collision operator is added. –Joke137 15:48, 9 Jun 2005 (UTC)

I have removed a couple of typos - in the equation of continuity, (which says nothing about convective derivatives being zero), and the role of N. Added a brief reference to the 'fluids analogy' (which is essentially what your proof is) because it's easier not to make mistakes (for me anyway) that way. And also on the motion of of a cloud of points, because I think it gives a feel for what's going on in phase space.

Linuxlad 14:18, 5 Jun 2005 (UTC)

Yes, thanks for this. The total derivative popped up in the equation of continuity – I must have accidentially copied it from the Liouville equaton. –Joke137 03:30, 6 Jun 2005 (UTC)


joke137 - I'm sorry I don't agree with your last few points - the phase space is system points of potentially quite complex entities - any thing you can write a Hamiltonian for. (Eg a million flying bedsteads tied together with string) Mixing of variables is by the by - the cancellation clearly arises from the relation between pairs of ps & qs.

Classically, the use of the theorem is to justify some version of the ergodic hypothesis ie in a time-stationary system the probability of a system being in a small phase space volume is proportional to the phase space volume. .

Lets take this off-line rather than slog it out here - my email is fizziks@n-cantrell.demon.co.uk Bob


[edit] category

IS this a MATHEMATICAL theorem? I think not. It's a theorem about the way the real world behaves. It may follow trivially from the property of symplectic spaces (as I'm always told), but you have to show a 1:1 correspondence between such spaces and real phase space coordinates - or do mathematicians assert that Hamilton's relations, or equivalently, Newton's laws, are a mathematical inevitability. If so, I would welcome a demonstration.:-) Bob aka Linuxlad 08:45, 7 November 2005 (UTC)

Yes, it's a mathematical theorem. The article needs modifications. The theorem state that the natural measure on the phase space(the Lebesuge measure, locally) is invariant under the Hamiltonian flow. More general, it gives nec and suff conditions when a smooth measure on a mannifold is invariant under a flow governed by a system of differential equations. Mct mht 04:57, 4 April 2006 (UTC)

Sorry, beg to differ - it is a theorem about a property of the real world, like Newtonian mechanics, and as such needs to be understood by physicists, physical chemists and astronomers. The clearest explanations are found on these sites. Wikipedia needs to keep the need for a physically-based understtanding in mind, please. Bob aka Linuxlad 10:51, 4 April 2006 (UTC)

Sorry, beg to differ - them's there are fightin' words. To physicists, its a "real world theorem". To mathematicians, its an important theorem in the general study of differential equations in the large. Its not far away from Frobenius theorem, which connects up several huge areas of mathematics. Your POV is showing: you think the physics explanation is clear, presumably because you have a physics education. Your "simple" explanation would leave the math undergrad scratching their head going "what the heck is this S**T?" Its a different language. In the end, you must know both languages.linas 05:52, 13 June 2006 (UTC)

Returning to this after a long break - sorry, the theorem only holds in worlds where Hamilton's relations between the ps and qs hold, or equivalently where Newton's laws are true - its key insight is thus a physical one. Bob aka Linuxlad (talk) 20:50, 28 November 2007 (UTC)

[edit] clarification

from the article:

which was Gibbs's name for the theorem

err...what was? Deepak 00:38, 31 May 2006 (UTC)

'Conservation of density in phase (space)', as it says. Linuxlad 08:27, 31 May 2006 (UTC)

[edit] Liouville - Louiville

We should note somewhere (not least in the keywords!) that Joseph's name and the eponymous theorem are extensively spelled/misspelled as Louiville in the literature. This includes not only the scientific literature but even an item in the existing Wikipedia entry! (try searching the page). I assume that this is because the latter spelling is more natural (indeed I only assume that the former is actually correct!)

I think that would belong in the article on the man himself. Deepak 16:08, 31 May 2006 (UTC)

[edit] Noether's theorem

Curiously, the article now points out that the tuplet (\rho, \rho\dot{q}^i,\rho\dot{p}_i) is a conserved current; this is a corollary of the proof. But this implies that Noether's theorem applies. What, exactly, in this context, is the set of symmetries implied by Noether's theorem? This article should make this connection explicit; Noether's theorem is a major landmark in physics, so this is just begging the question. linas 05:37, 13 June 2006 (UTC)

Never mind. linas 14:17, 13 June 2006 (UTC)

Noether's Theorem states that symmetry implies conservation. In the article it is said "Equivalently, the existence of a conserved current implies, via Noether's theorem, the existence of a symmetry. The symmetry is invariance under time translations, and the generator (or Noether charge) of the symmetry is the Hamiltonian." This needs to be changed, conserved quantities do not, at least by Noether's Theorem, imply symmetry. I'm guessing this should read something like " The symmetry of invariance under time translations implies, via Noether's Theorem, that current is conserved." However, since I'm not sure this is true, I didn't want to change this myself.

208.124.63.250 (talk) 14:03, 7 April 2008 (UTC)


[edit] Probable false claim requires proof

I believe the following claim to be false:

It is straightforward to show that as the cloud stretches in one coordinate – pi say – it shrinks in the corresponding qi direction

This is tantamount to saying that the diffeomorphism induced by every Hamiltonian flow preserves the symplectic leaves of the canonical coordinates. It's total phase volume that's preserved, not phase volume projected down to individual leaves. The condition stated is sufficient, but it's certainly not necessary. It might even be worthwhile to state a counterexample. A four-dimensional phase space is minimal. You can have Hamiltonians both where q_0,p_0 \rightarrow 0 and where q_0,q_1\rightarrow 0. Anybody want to illustrate the Hamiltonian functions? --Eh9 (talk) 19:12, 21 March 2008 (UTC)

The relationship holds because Hamilton's relations apply to pi, qi pairs, and so this ensures the ΔpiΔqi products (no summation convention) stay constant as the volume flows through phase space. Go back to say March 2005, or look on the web - not anything clever, but something for us simple physicists, or chemists. Bob aka Linuxlad (talk) 20:17, 21 March 2008 (UTC)
The double-linked pendulum provides a counterexample. In general, there are always periods of counter-phase amplitude oscillations in each pendulum as seen against the other. This corresponds to an quasi-oscillating transfer of phase volume between the two.
Mathematically, if all the mixed partials of the Hamiltonian are zero, that is, if \tfrac{\partial^2 H}{\partial q_i \partial q_j} = \tfrac{\partial^2 H}{\partial q_i \partial p_j} = \tfrac{\partial^2 H}{\partial p_i \partial p_j} = 0 when i \neq j , then the Hamiltonian can be expressed separably on 2-dimensional symplectic leaves, that is, H = \sum_{k=0}^{n}H_k(q_k,p_k) . In this case phase volume is preserved in leaves. If not, however, the mixed partial derivatives mean that cross-leaf dependencies appear in Hamilton's equations.--Eh9 (talk)
There's already a Wikipedia article on the Double pendulum.--Eh9 (talk) 02:51, 24 March 2008 (UTC)
Ta - (Lets continue the discussion of the double pendulum on your user page first). On the main issues, as I've written on your userpage, I think the statement on dpi v dqi is true to first order, for small times, for the faces of a suitably aligned small hypercube in phase space. Which is all that's really being claimed. The longer term gyrations and twists are by the by. Bob aka Linuxlad (talk) 08:47, 24 March 2008 (UTC)

Footnote - we've been here before, in both my physicists' 'baby notation' and Penrose's 2-forms - see March 2005 versions and p484 in the Penrose tome. Bob aka Linuxlad (talk) 13:58, 26 March 2008 (UTC)

  • There's a good treatment of Liouville's theorem in V.I.Arnold's Mathematical Methods of Classical Mechanics (in the 2nd edition, it's in section 16). It does not use differential forms but only basic calculus. The version with the Hamiltonian flow and the canonical symplectic form comes later in that book. The language of this proof can be extended to provide a disproof of the challenged assertion.
  • Penrose simply used 2-forms. Élie_Cartan invented differential forms, and, as I understand it, he did so for Einstein to do GR with (although I don't believe he ever switched), and, again as I recall, this was in the 1920's. Darboux's theorem in its original form is from 1882, and recognized as a 2n-form shortly after those objects were invented. It's not like this stuff is particularly recent. —Preceding unsigned comment added by Eh9 (talk • contribs) 15:52, 27 March 2008 (UTC)

[edit] Missing premise

The Liouville equation as stated is only true when \frac{\partial H}{\partial t} = 0.--Eh9 (talk) 11:28, 26 March 2008 (UTC)